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2017美赛A题O奖论文.pdf
For office use only
T1 ________________
T2 ________________
Team Control Number
55069
For office use only
F1 ________________
F2 ________________
T3 ________________
Problem Chosen
F3 ________________
T4 ________________
A
F4 ________________
2017 MCM/ICM Summary Sheet
The Rehabilitation of the Kariba Dam
Recently, the Institute of Risk Management of South Africa has just warned that
the Kariba dam is in desperate need of rehabilitation, otherwise the whole dam would
collapse, putting 3.5 million people at risk. Aimed to look for the best strategy with the
three options listed to maintain the dam, we employ AHP model to filter factors and
determine two most influential criteria, including potential costs and benefits. With the
weight of each criterion worked out, our model demonstrates that option 3 is the
optimal choice.
According to our choice, we are required to offer the recommendation as to the
number and placement of the new dams. Regarding it as a set covering problem, we
develop a multi-objective optimization model to minimize the number of smaller
dams while improving the water resources management capacity. Applying TOPSIS
evaluation method to get the demand of the electricity and water, we solve this problem
with genetic algorithm and get an approximate optimal solution with 12 smaller dams
and determine the location of them.
Taking the strategy for modulating the water flow into account, we construct a
joint operation of dam system to simulate the relationship among the smaller dams
with genetic algorithm approach. We define four kinds of year based on the Kariba’s
climate data of climate, namely, normal flow year, low flow year, high flow year and
differential year. Finally, these statistics could help us simulate the water flow of each
month in one year, then we obtain the water resources planning and modulating strategy.
The sensitivity analysis of our model has pointed out that small alteration in our
constraints (including removing an important city of the countries and changing the
measurement of the economic development index etc.) affects the location of some of
our dams slightly while the number of dams remains the same. Also we find that the
output coefficient is not an important factor for joint operation of the dam system, for
the reason that the discharge index and the capacity index would not change a lot with
the output coefficient changing.
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Contents
1 Overview ................................................................................................................... 1
1.1 Background ................................................................................................. 1
1.2 Restatement of the Problem ..................................................................... 1
1.3 Literature Review ....................................................................................... 2
2 Assumptions and Justifications ........................................................................ 2
3 Notation .................................................................................................................... 3
4 Model Overview ...................................................................................................... 4
5 Model Theory........................................................................................................... 5
5.1 Determination of the Number and Location of the Dams .................... 5
5.2 Joint operation of dam system model ..................................................... 9
6 Model Implementation and Results ................................................................ 11
6.1 The Number and Location ...................................................................... 11
6.2 The Strategy of Modulating Water Flow ............................................... 14
7 Sensitivity Analysis ............................................................................................. 15
7.1 The Model of Determination of the Number and Location ................. 15
7.2 The Model of Modulating Water Flow ................................................... 17
8 Further discussion .............................................................................................. 17
9 Strengths and Weaknesses .............................................................................. 18
9.1 Strengths ................................................................................................... 18
9.2 Weaknesses .............................................................................................. 19
10 Conclusion .......................................................................................................... 19
11 The Evaluation of Three Options ................................................................... 20
11.1 Establish a Hierarchical Model ............................................................. 20
11.2 Analysis and Results ............................................................................. 21
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1 Overview
1.1 Background
A Zambezi River Authority conference was held in March 2014, engineers warned that the
foundations of the dam had weakened and there was a possibility of dam failure unless repairs
were made.
On 3 October 2014 the BBC reported that “The Kariba Dam is in a dangerous state. Opened in
1959, it was built on a seemingly solid bed of basalt. However, in the past 50 years, the torrents
from the spillway have eroded that bedrock, carving a vast crater that has undercut the dam’s
foundations, engineers are now warning that without urgent repairs, the whole dam will collapse.
If that happened, a tsunami-like wall of water would rip through the Zambezi valley, reaching
the Mozambique border within eight hours. The torrent would overwhelm Mozambique’s
Cahora Bassa Dam and knock out 40% of southern Africa’s hydroelectric capacity. Along with
the devastation of wildlife in the valley, the Zambezi River Authority estimates that the live of
3.5 million people are at risk.”
On February 2015, Engineers have started on a R3.3bn rescue marathon to prevent the
“catastrophic failure” of the Kariba Dam. According to a World Bank special report on the
beleaguered structure-one of the biggest man-made dams in the world-a potential wall collapse
threatens the lives of about 3-million people living on the Zambezi River floodplain between
the hydro scheme on the Zambia-Zimbabwe border and the Mozambique coast. [1]
1.2 Restatement of the Problem
We are required to provide an overview of potential costs and benefits with the three options
already listed. Then we need to establish a model to determine the number and placement of
the new dams when removing the Kariba dam along the Zambezi River. The same overall water
management capabilities are also needed. In addition, we should consider emergency water
flow situations and restrictions regarding the locations and time, so that we could give out the
strategy for modulating the water flow through our new multiple dam system.
In order to solve those problems, we will proceed as follows:
Build a model to determine the number and location of the multiple dams.
Give the corresponding strategy of modulating water flow in different conditions.
In our model, we first establish a multi-objective model and use genetic algorithm determine
the number and location of the multiple dams. There are two goals improving the water
resources management capacity and reducing the cost. Besides, we add some constraints such
as water balance, water level, safety and water protection. We choose twenty suitable dam sites
and employ the genetic algorithm to solve the optimal problem to determine the number and
the location.
After determining the number and location of the dams, we construct our joint operation of dam
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system model and employ the genetic algorithm to solve the problem based on the thought of
dynamic programming. According to the Kariba’s climate data for about 30 years, we abstract
normal flow year, low flow year, high flow year and differential year. We use them to work out
the water resources planning and scheduling strategy. The construction of the discharge index
and the capacity index benefits an analysis and evaluation for joint operation of the dam
system’s performance in different month and year.
1.3 Literature Review
Dating back to 2004, the United States removed 72 dams in total, which created a historical
record. Therefore, it is high time for us to focus on the construction of dams concerning their
number and placement. Plenty of researchers have already made a number of notable papers to
address these problems
Alfer Weber (1909) first proposed a framework for location problem, which is an allocation
question with respect to space resource. Among the three classical location model, set covering
problem is a significant branch of siting issues. They explored a multi-objective location model
to tackle problems with siting optimal points. In their model, maximizing coverage rate in order
to satisfy every place’s need is the target function, the concentric point and the capacity
restrictions are constraint conditions. Thus, they could convert the optimization problem to the
mixed integer linear programming question.
After the set covering model was established, we can optimize our choice of siting the dams.
Then several scientists were devoted to building an optimal operation model to provide a
reasonable balance between safety and balance. They begin to figure out how the multiple dam
system would benefit or affect each other within its system. Masse(1940) first illustrated the
concept of it; their main computing method was to optimize water modulating strategy during
dispatching period.
Further studies are carried out about different methods to investigate the optimal operation
model, including dynamic programming algorithm and neural network algorithm based on
improved computer technique. Also, there is much theoretical analysis about location problem
since 1990. John Current and Morton O'Kelly(1992) suggested using a modified version of the
set covering location model, which still didn’t take the reality into account.
2 Assumptions and Justifications
To simplify our problems, we make the following basic assumptions, each of which is properly
justified.
The dam system is built downstream in the valley of the Kariba Dam. Because it’s
more convenient to build and also with less cost, which can be easily implemented.
The cost of the dam is mostly the same. Owing to the fact that the length of the canyon
is not large（24km）,geological conditions and climate conditions are mostly the same.
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Each dam’s water supply is mostly the same. Taking into account of safe operation of
the entire multi-dam system, we should make the burden of each dam to be the same as
much as possible.
The water quality of the dam system is the average of the water quality between the
two reservoirs. The river is flowing, so the water quality is mostly similar.
Water of the dams downstream only comes from dams upstream and natural
precipitation. According to Google Maps, there are no tributaries near the canyon. Also
with the principle of conservation of water, the formula should be maintained.
3 Notation
Abbreviation
Description
The set of cities
The set of dams
The distance from the
city to the nearest smaller dam
The electricity demand of
city
The water demand of
city
The discharge amount of the
dam at the end of period t,
The total amount of water released during period t
The amount of natural water in period t
The volume in the
period of the
dam
The inflow in the
period of the
dam
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Team # 55069 Page 4 of 28
The discharge volume in the
period of the
dam
The runoff volume between the
dam and the
dam
The actual water supply
dam in the
period
The planned water supply of the
dam in the
period
The hydraulic head of
dam in the
period
The fore-voltage factor of the
dam
The discharge flow indicators in the
period
The storage capacity indicators in the
period
4 Model Overview
To provide a detailed analysis of the option (3)-removing the Kariba dam and newly building
ten to twenty smaller dams. We need to determine the number and location of the multiple dams
first. And on that basis we must establish a model to modulate the water flow through the
multiple dam system to adapt to different situations. A reasonable balance between safety and
cost of our strategy is also needed.
Our first model allows us to determine the number and the location of the multiple dams. We
regard the optimal problem as a set covering problem and establish a multi-objective model to
solve the problem. There are two goals, namely improving water resources management
capacity and reducing the cost. And there are also some constraints including water balance,
water level, safety, water resources protection and number constraints. On the account that the
optimal problem is difficult to solve in polynomial time, so we use genetic algorithm to get the
solution.
After determining the numbers and the location, we establish a joint operation of dam system
model to gain a strategy about modulating water flow in different condition. Though it’s also a
multi-objective problem, it is different from the previous model. We use the maximization of
economic and social benefits as the objective and set some constraints, such as water balance,
reservoir capacity and discharge flow constraints. We use genetic algorithm to get the
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modulating strategy in different conditions.
In conclusion, we use programming and heuristic algorithm to solve the problem of building
dams and the modulating strategy. It’s relatively easy to achieve and it has a significant
guidance for the reality.
5 Model Theory
5.1 Determination of the Number and Location of the Dams
5.1.1 Establishment of the model
The construction of dams needs to consider many aspects, while at the same time it is subject
to economic, social, environmental constraints and other factors. In order to obtain the proper
number of dams and their location, we establish a multi-objective model.
The Objective
Improve Water Resources Management Capacity
The purpose of building smaller dams instead of larger dams is to manage water resources better,
mainly to satisfy dweller’s demand for the electricity consumption and water consumption
(Including agricultural, industrial and domestic water) of the neighboring cities. Demand may
vary between cities, but it is clear that for cities with greater demand, the dam should be built
closer to them. so we get that
Reduce the Cost
On the basis of ensuring the water supply and power supply, we should minimize the cost of
our plan. The whole cost consists of the removal of the Kariba dam and building new smaller
dams. Since the cost of removal is fixed, so we only consider the variable cost(building cost),
which is only related to the number of dams. So we should minimize the quantity of the smaller
dams.
The Constraints
Water Balance
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Team # 55069 Page 6 of 28
，
and
represent the discharge amount of the
dam at the end of period
t, the total amount of water released during period t and the amount of natural water in period t
respectively.
denotes the set of all higher dams of
dam. If
is empty, the
corresponding summation is zero.[2]
Water Level
The water level in the dam area should be kept between the dead water level and the limited
water level in the flood season. Dead Water Level, namely the lowest water level that allows a
reservoir to fall off under normal operating conditions. Flood limited water level is the
requirements of control over flood to limit reservoirs’ water storage.
Safety
The construction of multiple smaller dams is at least as safe as the original larger dams. The
safety considerations for dam construction mainly include reducing the probability of dam
failure, thereby reducing damage to dams downstream and enhancing resistance ability to
extreme weather.
represents the safety of multiple dams system, while
denotes the safety of
the existing Kariba dam.
Water Resources Protection
The construction of smaller dams should require a higher degree of protection for the water
resources as the replacement of the existing dam.
represents the environment protection of the multiple dams system, while
denotes the environment protection of the existing Kariba dam.
Number
The number of small dams should be greater than ten and less than twenty. Option 3 is replacing
the Kariba dam with a series of ten to twenty smaller dams.
To ensure the continuity of the model establishment, the parameters of the constraints will be
described in the next part. And in the sixth section, we will clarify how to calculate the demand
of electricity(
),the demand of the water(
),the safety of different dams and the
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